#### Vol. 1, No. 1, 2013

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On the density of abelian surfaces with Tate-Shafarevich group of order five times a square

### Stefan Keil and Remke Kloosterman

Vol. 1 (2013), No. 1, 413–435
##### Abstract

Let $A={E}_{\mathfrak{1}}×{E}_{\mathfrak{2}}$ be the product of two elliptic curves over $ℚ$, each having a rational $\mathfrak{5}$-torsion point ${P}_{i}$. Set $B:=A∕〈\left({P}_{\mathfrak{1}},{P}_{\mathfrak{2}}\right)〉$. In this paper we give an algorithm to decide whether the order of the Tate-Shafarevich group of the abelian surface $B$ is square or five times a square, under the assumptions that we can find a basis for the Mordell-Weil groups of ${E}_{\mathfrak{1}}$ and ${E}_{\mathfrak{2}}$ and that the Tate-Shafarevich groups of ${E}_{\mathfrak{1}}$ and ${E}_{\mathfrak{2}}$ are finite.

We considered all pairs $\left({E}_{\mathfrak{1}},{E}_{\mathfrak{2}}\right)$ with prescribed bounds on the conductor and the coefficients in a minimal Weierstrass equation. In total we considered around $\mathfrak{2}\mathfrak{0}.\mathfrak{0}$ million abelian surfaces, of which $\mathfrak{4}\mathfrak{9}.\mathfrak{1}\mathfrak{6}%$ have Tate-Shafarevich groups of nonsquare order.

##### Keywords
Tate-Shafarevich groups, abelian surface, Cassels-Tate equation
##### Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 11G40, 14G10, 14K15
##### Milestones
Published: 14 November 2013
##### Authors
 Stefan Keil Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 D-10099 Berlin Germany Remke Kloosterman Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 D-10099 Berlin Germany